lab manual xi (setting (hindi) on13-08-09) 11 20ncert.nic.in/ncerts/l/khlm402.pdf · tslk vko`qfr...

xf.kr 41

jpuk dh fof/

1- mi;qDr vkdkj osQ dkMZcksMZ ij pkVZ isij fpidkb,A

2- fcanq O ij dkVrh gqbZ nks yacor~ js[kk,¡ [khafp,A (nsf[k, vko`Qfr 11)

3- ,dd yackbZ (eku yhft, 10 cm) dk ,d Mksjk yhft, tks OX osQ vuqfn'k la[;k 1dks fu#fir djrk gSA Mksjs ds ,d fljs dks O ij rFkk nwljs fljs dks A ij fLFkj dhft,tSlk vko`Qfr (11) esa fn[kk;k x;k gSA

4- Mksjs osQ nwljs fljs tks A ij gS] dks eqDr dhft, vkSj Mksjs dks 90º, 180º, 270º vkSj360º ij ?kqekb, vkSj eqDr fljs dh fofHkUu fLFkfr;ksa dks Øe'k% A

1, A

2,A

3 vksj A

4 ls

vafdr dhft, tSlk fd vko`Qfr esa fn[kk;k x;k gSA

mís'; vko';d lkexzh

–1i = vkSj blosQ iw.kk±dh; ?kkrksa

dh T;kferh; :i esa foospuk djukA

dkMZcksMZ] pkVZ isij] LosQp isu] :yj]ijdkj] xksan] dhysa] Mksjk] bR;kfnA

fØ;kdyki 11

04/26/2018

42 iz;ksx'kkyk iqfLrdk

izn'kZu

1- vkjxSaM+ ry esa OA, OA1, OA

2, OA

3, OA

4, Øe'k% 1, i,-1,-i, 1dks fu#fir djrs gSaA

2- OA1 = i = 1 × i, OA

2 = – 1 = i × i = i2, OA

3 = – i = i × i × i = i3 vkSj vkxs Hkh

blh izdkjA izR;sd ckj] OA dk 90° ls ?kw.kZu i ls xq.ku osQ rqY; gSA bl fy, i dks 90°

osQ ?kw.kZu dk xq.kkRed ?kVd dgrs gSaA

izs{k.k

1- OA dk 90º djus ij OA1 = 1 × i = _________

2- OA dk 180° ?kw.kZu u djus ij OA2 = 1 × __ × __ = ______

3- OA dk 270º (3 ledks.k) ?kw.kZu djus ij

OA3 = 1 × _______ × ______× ______ = ______

4- OA dk 360º (4 ledks.k) ?kw.kZu djus ij

OA4 = 1 × _______ × ______× ______ × ______ = __________

5- OA dk n-ledks.k ?kw.kZu djus ij

OAn = 1 × _______ × ______× ______ × ______ × ... n ckj = __________

vuqiz;ksx

bl fØ;kdyki osQ mi;ksx ls i dh fdlh Hkh iw.kk±d ?kkr dk eku Kkr fd;k tk ldrk gSA

04/26/2018

xf.kr 43

jpuk dh fof/

1- leku yackbZ osQ nks rkj yhft,A

2- budks ry osQ fcanq ^O* (IykbZoqM 'khV) ij ijLij yacor~ fLFkj dhft, rkfd os x-v{kvkSj y-v{k dks fu#fir djsaA (nsf[k, vko`Qfr 12)

3- ,d rkj dk VqdM+k yhft, vkSj mls bl izdkj fLFkj dhft, fd og x-v{k dh èkukRedfn'kk esa O ls a bdkbZ dh nwjh rFkk y-v{k dks O ls uhps dh vksj a bdkbZ dh nwjh ijfeys tSlk fd vko`Qfr esa fn[kk;k x;k gSA bu fcanqvksa dks Øe'k% B vkSj A ls vafdrdhft,A

mís'; vko';d lkexzhjSf[kd iQyuksa dh lgk;rk ls vkjs[k fof/}kjk ,d f}?kkrh; iQyu dks izkIr djukA

IykbZoqM 'khV] rkjksa osQ VqdM+s

fØ;kdyki 12

04/26/2018


4- blhizdkj nwljk rkj ysdj mls bl izdkj fLFkj dhft, fd og x-v{k dh èkukRed fn'kkesa 0 ls b bdkbZ dh nwjh rFkk y-v{k dks O ls uhps dh vksj b bdkbZ dh nwjh ij feystSlk fd vko`Qfr 12 esa fn[kk;k x;k gSA bu fcanqvksa dks Øe'k% D vkSj C ls vafdrdhft,A

5- ,d vU; rkj yhft, vkSj bldks bl izdkj fLFkj dhft, fd ;g mu fcanqvksa ls gksdjtk, tg¡k lh/s rkj x-v{k ij feyrs gSa ;g rkj ,d oØ (ijoy;) osQ vkdkj dk gStSlk fd vko`Qfr 12 esa fn[kk;k x;k gSA

izn'kZu

1- fcanqvksa A vkSj B ls tkus okyk rkj ,d ljy js[kk dks fu#fir djrk gS ftls y = x – a }kjkfn;k tkrk gSA ;g x-v{k vkSj y-v{k dk Øe'k% (a, 0) vkSj (0, – a) ij izfrPNsn djrkgSA

2- fcanqvks C vkSj D ls tkus okyk rkj ,d ljy js[kk dks fu#fir djrk gS ftls y = x– b }kjkfn;k tkrk gSA ;g x v{k vkSj y v{k dk Øe'k% (b, 0) vkSj (0, – b) ij izfrPNsn djrk gSA

3- fcanqvksa B vkSj D ls tkus okyk rkj ,d oØ dks fu#fir djrk gS ftls iQyuy = k (x–a) (x–b) = k [x2 – (a+b) x + ab], tgk¡ k ,d LosPN vpj gS] }kjk fn;ktkrk gSA

izs{k.k

1- jSf[kd iQyu y = x – a }kjk nh xbZ js[kk x-v{k dks fcanq ______ ij izfrPNsn djrhgS ftlosQ funsZ'kkad _________ gSaA

2- jSf[kd iQyu y = x – b }kjk nh xbZ js[kk x-v{k dks fcanq ______ ij izfrPNsn djrhgS ftlosQ funsZ'kkad ______ gSaA

3- fcanqvksa C vkSj D ls tkus okyk oØ] iQyu y = ______ }kjk fn;k tkrk gS tks ,d______ iQyu gSA

vuqiz;ksx

;g fØ;kdyki f}?kkrh; iQyuksa osQ 'kwU;ksa rFkk muosQ xzkI+kQ osQ vkdkj dks le>us esa lgk;d gSA

04/26/2018

xf.kr 45

jpuk dh fofèk

1- mi;qDr vkdkj dk ,d dkMZcksMZ yhft, vkSj ml ij ,d lI+ksQn dkx”k fpidkb,A

2- x-v{k vkSj y-v{k dks fu#fir djus okyh nks ijLij yac js[kk,¡ X OX Y OY vkjS[khafp,A

3- nh xbZ jSf[kd vlfedk ds laxr jSf[kd lehdj.k dk vkys[k [khfp,A

4- nksuksa vèkZ leryksa dks I vkSj II ls vafdr dhft, tSlk vko`Qfr 13 esa fn[kk;k x;k gSA

mís'; vko';d lkexzh;g lR;kfir djuk fd ,d nh xbZvlfedk eku yhft, 5x + 4y – 40 < 0,

tks ax + by + c < 0, a, b > 0, c < 0

izdkj dh gS] nks vèkZ leryksa esa lsosQoy ,d dks fu#fir djrk gSA

dkMZcksMZ] lI+ksQn eksVk dkx”k] LosQp isu]:yj] xksan

fØ;kdyki 13

04/26/2018


izn'kZu

1- vèkZ lery I esa oqQN fcanq O(0, 0), A(1, 1), B(3, 2), C(4, 3), D(–1, –1) rFkk vèkZlery II esa oqQN fcanq E(4, 7), F(8, 4), G(9, 5), H(7, 5) vafdr dhft,A

2- (i) fcanq O (0,0) osQ funsZ'kkadksa dks vlfedk osQ oke i{k esa jf[k,A

oke i{k dk eku = 5 (0) + 4 (0) – 40 = – 40 < 0

bl izdkj O osQ funZs'kakd tks vèkZ lery I esa fLFkr gSa vlfedk dks larq"V djrs gSaSA

(ii) fcanq E (4, 7) osQ funsZa'kkadksa dks vlfedk ds oke i{k esa jf[k,A

oke i{k dk eku = 5(4) + 4(7) – 40 = 8 </ 0 vkSj bl izdkj fcanq E osQ funsZ'kakdtks vèkZ lery II esa fLFkr gSa] nh xbZ vlfedk dks larq"V ugha djrs gSaA

(iii)fcanq F (8, 4) osQ funsZ'kkadksa dks vlfedk ds oke i{k esa jf[k,A

oke i{k dk eku = 5(8) + 4(4) – 40 = 16 </ 0

bl izdkj fcanq F osQ funZs'kakd tks vèkZ lery II esa fLFkr gSa vlfedk dks larq"V ughadjrs gSaSA

(iv) fcanq C (4, 3) osQ funsZa'kkadksa dks vlfedk ds oke i{k esa jf[k,A

oke i{k dk eku = 5(4) + 4(3) – 40 = – 8 < 0

vr% fcanq C osQ funsZ'kkad tks vèkZ lery I esa fLFkr gS] vlfedk dks larq"V djrs gSaA

(v) fcanq D(–1, –1) osQ funsZ'kkadksa dks vlfedk ds oke i{k esa jf[k,A

oke i{k dk eku = 5(–1) + 4 (–) – 40 = – 49 < 0

vr% fcanq D osQ funsZ'kakd tks vèkZ lery I esa fLFkr gS] vlfedk dks larq"V djrs gSaA

(vi) blh izdkj fcanq A (1, 1), tks vèkZ lery I esa fLFkr gSa] vlfedk dks larq"V djrsgSaA fcanq G (9, 5), H (7, 5) tks vèkZ lery II esa fLFkr gS] vlfedk dks larq"V ughadjrs gSaA

04/26/2018

xf.kr 47

bl izdkj] lHkh fcanq O, A, B, C, tks jSf[kd vlfedk dks 5x + 4y – 40 < 0 larq"V djrsgSa osQoy vèkZ lery I esa fLFkr gSa rFkk lHkh fcanq E, F, G, H tks jSf[kd vlfedk dks larq"Vugha djrs gSa os vèkZ lery II esa fLFkr gSaA bl izdkj] nh xbZ vlfedk dk vkjs[k nks esa lsosQoy ,d laxr vèkZ lery dks fu#fir djrk gSA

izs{k.k

1- fcanq A ds funsZ'kkad nh xbZ vlfedk dks __________ djrs gSaA (larq"V@ larq"V ugha)

2- fcanq G ds funsZ'kkad nh xbZ vlfedk dks __________ djrs gSaA (larq"V@ larq"V ugha)

3- fcanq H ds funsZ'kkad nh xbZ vlfedk dks __________ djrs gSaA (larq"V@ larq"V ugha)

4- fcanq E ds funsZ'kkad nh xbZ vlfedk dks __________ djrs gSaA (larq"V@ larq"V ugha)

5- fcanq F ds funsZ'kkad nh xbZ vlfedk dks __________ djrs gSaA (larq"V@ larq"V ugha)

6- nh xbZ vlfedk dk vkys[k osQoy vèkZ lery __________ gSA

vuqiz;ksx

bl fØ;kdyki dk mi;ksx ml vèkZ lery dksfu/kZfjr djus esa fd;k tk ldrk gS tks nh xbZvlfedk dk gy gksrk gSA

fVIi.kh

bl fØ;kdyki dk ax + by + c > 0 osQizdkj dh vlfedkvksa osQ fy, Hkh fu"iknufd;k tk ldrk gSA

04/26/2018


jpuk dh fof/

1. ,d xÙks dh 'khV yhft, vkSj ml ij lI+ksQn dkx”k fpidkb,A

2. dkMZcksMZ ls mi;qDr lkb”k osQ 5 ,d tSls dkMZ dkfV,A

3. bu dkMksZ ij C1, C

2, C

3, C

4 vkSj C

5 vafdr dhft,A

izn'kZu

1. fn, x, ik¡p dkMZ esa ls ,d dkMZ dk p;u dhft,A

2. eku yhft, fd igyk p;fur dkMZ C1 gSA rc 'ks"k pkj dkMksZ esa ls vU; nks dkMZ C

2C

3,

C2C

4, C

2C

5, C

3C

4, C

3C

5 vkSj

C

4C

5 gks ldrs gSA bl izdkj] laHko p;u C

1C

2C

3 ,

C1C

2C

4, C

1C

2C

5,C

1C

3C

4, C

1C

3C

5, C

1C

4C

5 gSaA budks ,d dkx”k ij vafdr dhft,A

3. eku yhft, fd igyk p;fur dkMZ C2 gSA rc 'ks"k pkj dkMks± esa ls vU; nks dkMZ C

1C

3,

C1C

4, C

1C

5, C

3C

4, C

3C

5, C

4C

5 gks ldrs gSaA bl izdkj laHko p;u C

2C

1C

3 , C

2C

1C

4,

C2C

1C

5,C

2C

3C

4, C

2C

3C

5, C

2C

4C

5 gSaA budks mlh dkx”k ij vfdar dhft,A

4. eku yhft, fd igyk p;fur dkMZ C3 gSA rc 'ks"k pkj dkMksZa esa ls vU; nks dkMZ

C

1C

2,

C1C

4, C

1C

5, C

2C

4, C

2C

5, C

4C

5 gks ldrs gSA bl izdkj laHko p;u C

3C

1C

2, C

3C

1C

4,

C3C

1C

5,C

3C

2C

4, C

3C

2C

5, C

3C

4C

5 gSaA budks mlh dkx”k ij vafdr dhft,A

5. eku yhft, fd igyk p;fur dkMZ C4 gSA rc 'ks"k pkj dkMksZ es ls vU; nks dkMZ

C

1C

2,

C1C

3, C

2C

3, C

1C

5, C

2C

5, C

3C

5 gks ldrs gSaA bl izdkj] laHko p;u C

4C

1C

2,

C4C

1C

3,C

4C

2C

3, C

4C

1C

5, C

4C

2C

5, C

4C

3C


mís'; vko';d lkexzh;g Kkr djuk fd fn, x, ik¡p dkMksZa esa ls rhudkMksZa dk p;u fdrus izdkj ls fd;k tk ldrkgSA

dkMZcksMZ] lI+ksQn dkx”k dh 'khV] LosQpisu] dVj

fØ;kdyki 14

04/26/2018

xf.kr 49

6. eku yhft, fd igyk C5 gSaA rc 'ks"k pkj dkMksZa esa ls vU; nks dkMZ C

1C

2, C

1C

3, C

1C

4,

C2C

3, C

2C

4, C

3C

4 gks ldrs gSA bl izdkj laHko p;u C

5C

1C

2,C

5C

1C

3, C

5C

1C

4,

C5C

2C

3, C

5C

2C

4, C

5C

3C


7. vc ml dkxT+k ij è;ku nhft, ftlij laHkkfor p;uksa dks lwphc¼ fd;k FkkA ;gk¡ oqQy30 laHkkfor p;u gS vkSj izR;sd p;u dh rhu ckj iqujko`fÙk gqbZ gSA blfy, fHkUu p;uksadh la[;k 30 3 10= ÷ = gS tks fd 5C

3 osQ cjkcj gSA

izs{k.k

1. C1C

2C

3 , C

2C

1C

3 vkSj C

3C

1C

2 _______ p;u dks fUk#fir djrs gSaA

2. C1C

2C

4, _____________, ____________ ,d gh p;u dks fu#fir djrs gSA

3. C2C

1C

5 , C

1C

2C

5, C

1C

2C

3 esa ls ________ vkSj ________ ,d gh p;u dks

fu#fir djrs gSaA

4. C2C

1C

5 , C

1C

2C

3 _______ p;uksa dks fu#fir djrs gSaA

5. C3C

1C

5, C

1C

4C

3, C

5C

3C

4, C

4C

2C

5, C

2C

4C

3, C

1C

3C

5 esa ls

C3C

1C

5, ________ ,d gh p;u dks fUk:fir djrs gSaA

C3C

1C

5, C

1C

4C

3, ______, _____, vyx&vyx p;uksa dks fu#fir djrs gSaA

vuqiz;ksx

bl izdkj osQ fØ;kdyki dk mi;ksx nh xbZ n oLrqvksa esa ls r oLrqvksa dks pquus dh lEHkkfor

la[;k osQ O;kid lw=k vFkkZr] ( )

!C

! – !

nnr

r n r= dks le>us esa fd;k tk ldrk gSA

04/26/2018


jpuk dh fof/

1. ,d Mªkabx cksMZ yhft, vkSj ml ij lI+ksQn dkx”k fpidkb,A

2. oqQN ekfpl dh rhfy;k¡ yhft, vkSj mUgsa ,sls O;ofLFkr dhft,A tSlk fd vko`Qfr 15esa fn[kk;k x;k gSA

mís'; vko';d lkexzhikWLdy f=kHkqt dh jpuk djuk vkSj ,df}in dh nh xbZ /ukRed iw.kkZad ?kkr osQfy, f}in izlkj fy[kukA

Mªkbax cksMZ] lI+ksQn dkx”k] ekfpl dhrhfy;k¡] xksan

fØ;kdyki 15

04/26/2018

xf.kr 51

3. la[;kvksa dks fuEu izdkj fyf[k,µ

1 (igyh iafDr)

1 1 (nwljh iafDr)

1 2 1 (rhljh iafDr)

1 3 3 1 (pkSFkh iafDr), 1 4 6 4 1 (ik¡poha iafDr) vkSj blh izdkj vkxs Hkh

(nsf[k, vko`Qfr 15)

4. (a + b)n dk f}in izlkj fy[kus osQ fy,] (n + 1)th oha iafDr esa nh xbZ la[;kvksa dk iz;ksx djsaA

izn'kZu

1. miZ;qDr vkoQfr ,d f=kHkqt tSlh fn[krh gS vkSj bls ikLdy f=kHkqt osQ oxZ esa j[kk tkrk gSA

2. nwljh iafDr dh la[;k,¡ (a + b)1 osQ f}in izlkj osQ inksa osQ xq.kakd gSA rhljh iafDr dhla[;k,¡ (a + b)2 osQ f}in izlkj osQ inksa osQ xq.kakd gSa] pkSFkh iafDr dh la[;k,¡ (a + b)3

osQ f}in izlkj osQ inksa osQ xqa.kkd gSaA ik¡poha iafDr dh la[;k,¡ (a + b)4 osQ f}in izlkjosQ inksa osQ xq.kakd gSa vkSj blh izdkj vkxs HkhA

izs{k.k

1. ik¡poha iafDr dh la[;k,¡ _________ gSa tks _______osQ f}in izlkj osQ xq.kkad gSaA

2. lkroha iafDr dh la[;k,¡ _________ gSa tks _______osQ f}in izlkj osQ inksa osQxq.kkad gSaA

3. (a + b)3 = ___ a3 + ___a2b + ___ab2 + ___b3

4. (a + b)6 =___a6 +___a5b + ___a4b2 + ___a3b3 + ___a2b4 + ___ab5 + ___b6

5. (a + b)5 = ___ +___+ ___+ ___ + ___+ ___.

6. (a + b)8 = ___ +___ +___+ ___ + ___+ ___ + ___ + ___+ ___

7. (a + b)10 =___ + ___ + ___+ ___ + ___+ ___ + ___+___+ ___+ ___+ __

vuqiz;ksx

bl fØ;kdyki dk mi;ksx (a + b)n osQ f}in izlkj osQ fy, fd;k tk ldrk gSA tgk¡ n ,d/ukRed iw.kk±d gSA

04/26/2018


jpuk dh fof/

1. ydM+h ;k IykfLVd dk 1 ( = 12) ,dd ?ku yhft, tSlk fd vko`Qfr 16.1 esa fn[kk;kx;k gSA

2. ydM+h ;k IykfLVd osQ 4 ( = 22) ,dd ?ku yhft, vkSj ,d ?kukHk cukb, tSlk fdvko`Qfr 16.2 esa fn[kk;k x;k gSA

3. ydM+h ;k IykfLVd osQ 9 ( = 32) ,dd ?ku yhft, vkSj ,d ?kukHk cukb, tSlk fdvko`Qfr 16.3 esa fn[kk;k x;k gSA

4. ydM+h ;k IykfLVd osQ 16 (= 42) ,dd ?ku yhft, vkSj ,d ?kukHk cukb, tSlk fdvko`Qfr 16.4 esa fn[kk;k x;k gSA

5. vko`Qfr 16.1 ls 16.4 rd osQ lHkh ?ku ,ao ?kukHkksa dks bl izdkj O;ofLFkr dhft, fdos ,d ,s'ksyku (echelon) izdkj dh lajpuk djsa tSlk fd vko`Qfr 16.5 esa fn[kk;kx;k gSA

mís'; vko';d lkexzhizFke n izko`Qr la[;kvksa osQ ;ksx dk lw=k KkrdjukA

ydM+h ;k IykfLVd osQ ,dd ?ku] jaxhudkx”k] xksan vkSj dhysaA

fØ;kdyki 16

04/26/2018

xf.kr 53

6. bl izdkj osQ N% ,s'ksyku izdkj dh lajpuk dhft,A buesa ls ,d dks vko`Qfr 16.5 esafn[kk;k x;k gSA

7. bu N% lajpukvksa dks ,d foLr`r ?kukHk (cuboidal block) cukus osQ fy, O;ofLFkrdhft, tSlk fd vko`Qfr 16.6 esa fn[kk;k x;k gSA

izn'kZu

1. vko`Qfr 16.5 esa nh xbZ lajpuk dk vk;ru = (1 + 4 + 9 + 16) ?ku bdkbZ

= (12 + 22 + 32 + 42) ?ku bdkbZ gSA

2. bl izdkj dh 6 lajpukvksa dk vk;ru = 6 (12 + 22 + 32 + 42) ?ku bdkbZ gSA

3. vko`Qfr 16.6 esa cus ?kukHk Cykd dk vk;ru (tks ,d ,d ,slk ?kukHk gS ftldhfoHkk,¡ = 4 × 5 × 9 ) = 4 (4 + 1) (2 × 4 + 1) gSA

4. bl izdkj] 6 (12 + 22 + 32 + 42) = 4 × (4 + 1) × (2 × 4 + 1)

vFkkZr~ 12 + 22 + 32 + 42 = 1

6 [4 × (4 + 1) × (2 × 4 + 1)]

izs{k.k

1. 12 + 22 + 32 + 42 = 1

6 ( _____ ) × ( _____ ) × ( _____ ).

04/26/2018


2. 12 + 22 + 32 + 42 + 52 = 1

6 ( _____ ) × ( _____ ) × ( _____ ).

3. 12 + 22 + 32 + 42 + ... + 102 = 1

6 ( _____ ) × ( _____ ) × ( _____ ).

4. 12 + 22 + 32 + 42 ... + 252 = 1

6 ( _____ ) × ( _____ ) × ( _____ ).

5. 12 + 22 + 32 + 42 ... + 1002 = 1

6 ( _____ ) × ( _____ ) × ( _____ ).

vuqiz;ksx

bl fØ;kdyki dk iz;ksx izFke n izko`Qr la[;kvksa osQ oxksZa dk ;ksx izkIr djus osQ fy, fd;ktk ldrk gS tks bl izdkj gS

12 + 22 + 32 + ... + n2 = 1

6n (n + 1) (2n + 1)

04/26/2018

xf.kr 55

jpuk dh fof/

1. vko`Qfr 17.1 esa fn[kk, x, vuqlkj 1, 4, 9, 16, 25 ... ,dd oxZ yhft, vkSj lHkh dks,d jax tSls dkys jax ls Hkfj,A

mís'; vko';d lkexzhizFke n izko`Qr la[;kvksa osQ ;ksx dk lw=k izkIrdjus dh oSdfYid fof/A

ydM+h ;k IykfLVd osQ ,dd oxZ] jaxhuisaflyas ;k LosQp isu] :yj

fØ;kdyki 17

04/26/2018


2. 1, 4, 9, 16, 25 ... ,dd oxksZ dk ,d vU; leqPp; yhft, tSlk vko`Qfr 17.2 esafn[kk;k x;k gS vkSj lHkh esa ,d jax tSls gjk jax Hkfj,A

3. 1, 4, 9, 16, 25 ... ,dd oxksZ dk ,d rhljk leqPp; yhft, tSlk vko`Qfr 17.3 esafn[kk;k x;k gS vkSj bu ,dd oxksZ dks vyx&vyx jxksa ls Hkfj,A

4. ,dd oxks± osQ bu rhuksa leqPp;ksa dks ,d vk;r osQ :i esa O;ofLFkr dhft, tSlkvko`Qfr 17.4 esa fn[kk;k x;k gSA

izn'kZu

1. vko`Qfr 17.1 esa fn, x, ,d leqPp; dk {ks=kiQy gS&

(1 + 4 + 9 + 16 + 25) oxZ bdkbZ

= (12 + 22 + 32 + 42 + 52) oxZ bdkbZ2. ,sls 3 leqPp;ksa dk {ks=kiQy = 3 (12 + 22 + 32 + 42 + 52)A

04/26/2018

xf.kr 57

3. vk;r dk {ks=kiQy = 15 × 11 = 5 6

2

×

[2 (5) + 1]

blfy, 3 (12 + 22 + 32 + 42 + 52) = 1

2 [5 × 6] [2 (5) + 1]

;k 12 + 22 + 32 + 42 + 52 = 1

6 [5 × (5 + 1)] [2 (5) + 1].

izsz{k.k

3 (12 + 22 + 32 + 42 + 52) = 1

2 ( ____ × ____) ( ___ + 1)

⇒ 12 + 22 + 32 + 42 + 52 = 1

6 ( ____ × ____) ( ___ + 1)

blfy,] 12 + 22 + 32 + 42 + 52 + 62 + 72 = 1

6 ( ____ × ____) ( ___ + 1)

12 + 22 + 32 + 42 +...+ 102 = 1

6 ( ____ × ____) ( ___ + 1).

vuqiz;ksx

bl fØ;kdyki dk iz;ksx

12 + 22 + 32 + ... + n2 = 1

6 (n × (n + 1) (2n + 1) dks izekf.kr djus osQ fy,

fd;k tk ldrk gSA

04/26/2018


jpuk dh fof/

1. pkVZ isij ls a × b (a > b) foekvksa okys pkj vk;rkdkj VqdM+s dkfV,A

2. vko`Qfr 18 dh Hkk¡fr pkjks vk;rkdkj VqdM+ksa dks O;ofLFkr dhft,A

izn'kZu

1. ABCD ,d oxZ gS ftldh Hkqtk (a + b) bdkbZ gSA

2. oxZ ABCD dk {ks=kiQy = (a + b)2 oxZ bdkbZ

3. pkj vk;rkdkj VqdM+ksa dk {ks=kiQy = 4 (ab) = 4ab oxZ bdkbZ

mís'; vko';d lkexzh;g iznf'kZr djuk fd nks fofHkUu ?kukRedla[;kvksa dk lekarj ekè;] xq.kksÙkj Ekk?; lslnSo vf/d gksrk gSA

jaxhu pkVZ isij] LosQy] LosQp isu] dVj

fØ;kdyki 18

04/26/2018

xf.kr 59

4. PQRS ,d oxZ gS ftldh Hkqtk (a – b) bdkbZ gSA

5. (ABCD) dk {ks=kiQy = pkj vk;rkdkj VqdM+ksa dk {ks=kiQy + oxZ PQRS dk {ks=kiQy

∴ (ABCD) dk {ks=kiQy > pkj vk;rkdkj VqdM+ksa osQ {ks=kiQy dk ;ksx

vFkkZr~ (a + b)2 > 4 ab

;k2

2

a b

+ > ab

∴2

a b+ > ab vFkkZr~ A.M. > G.M.

izs{k.k

a = 5cm, b = 3cm yhft,

∴ AB = a + b = ________ bdkbZ

ABCD dk {ks=kiQy = (a + b)2 = _______ oxZ bdkbZ

izR;sd vk;r dk {ks=kiQy = ab = _______ oxZ bdkbZ

oxZ PQRS dk {ks=kiQy = (a – b)2 = _________ oxZ bdkbZ

ABCD = 4 (vk;rkdkj VqdM+ksa dk {ks=kiQy) + ( PQRS dk {ks=kiQy)

_______ = 4 ( _______ ) + ( _______ )

∴ ________ > 4 ( _______ )

vFkkZr~ (a + b)2 > 4 ab ;k 2

2

a b

+ > ab

;k 2

a b+ ab> blfy, AM > GM

04/26/2018


jpuk dh fof/

1. FkeksZdksy dh mi;qDr oxkZdkj 'khV yhft, vkSj mlosQ mQij pkVZ isij fpidkb,A

2. fpiosQ gq, pkVZ isij ij {kSfrt vkSj mQèokZ/j js[kk,¡ bl izdkj cukb, fd 225 NksVs oxZcu tk,¡ tSlk vko`Qfr 19 esa fn[kk;k x;k gSA

3. lcls mQij ckb± vksj osQ oxZ ij ,d FkeksZdksy dh xsan dks fiu dh lgk;rk ls fLFkj djsa

mís'; vko';d lkexzhizFke n izko`Qr la[;kvksa osQ ?kuksa osQ ;ksx osQlw=k dks LFkkfir djukA

FkeksZdksy 'khV] FkeksZdksy dh xssansa] fiu]isafly] :yj] xksan] pkVZ isij] dVj bR;kfnA

fØ;kdyki 19

4. mlh oxkZdkj 'khV ij 23 vFkkZr~ 8 FkeksZdksy osQ xksys 8 fiuksa dh lgk;rk ls igys oxZ osQlayXu fLFkj djsa tSlk vko`Qfr esa fn[kk;k x;k gSA

04/26/2018

xf.kr 61

5. mlh oxkZdkj 'khV ij 27 fiuksa dh lgk;rk ls FkeksZdksy dh 33 vFkkZr~ 27 xsnksa dks igys8 oxksZ osQ layXu fLFkj djsaA

6. blh izdkj FkeksZdksy dh xsnksa dks rc rd fLFkj djrs jgsa tc rd lHkh oxZ Hkj u tk,¡(vko`Qfr 19.1 nsf[k,)

izn'kZu

1. layXud I esa xasnksa dh la[;k 2

3 1 21 1

2

× = = =

2. layXud II esa xsanksa dh la[;k 3 31 2 9= + =

22 3

2

× =

3. layXud III esa xasnksa dh la[;k = 13 + 23 + 33 = 36

23 4

2

× =

4. layXud IV esa xsanksas dh la[;k = 13 + 23 + 33 + 43 = 100

24 5

2

× =

5. layXud V esa xasnksa dh la[;k = 13 + 23 + 43 + 53

25 6

2

× =

izs{k.k

xsanksa dh okLrfod fxurh djus ijµ

1. layXud I esa xsanksa dh la[;k 2

3 1 21 _______

2

× = = =

2. layXud II esa xsanksa dh la[;k

2

3 3 ___ ___1 2 _______

____

×= + = =

04/26/2018


3. layXud III esa xsanksa dh la[;k

2

3 3 ___ ___1 2 _______

____

×= + = =

-

4. layXud IV esa xsanksa dh la[;k 2

3 3 ___ ___1 2 _______ _______

2

× = + + = =

-

5. layXud V esa xsanksa dh la[;k

( ) ( ) ( ) ( ) ( )2

3 3 3 3 3 ___ _____ __ __ __ __ ___

2

× = + + + + = =

-

vuqiz;ksx

bl ifj.kke dk mi;ksx izFke n izko`Qr la[;kvksa osQ ?kuksa dk ;ksx vFkkZr~

13 + 23 + 23 + ... + n3 =

2( 1)

2

n n +

Kkr djus osQ fy, fd;k tk ldrk gSA

04/26/2018

xf.kr 63

jpuk dh fof/

1. mi;qDr vkdkj dk ,d xÙkk yhft, vkSj ml ij lI+ksQn dkx”k fpidkb,A

2. lI+ksQn dkx”k ij nks yacor~ js[kk¡, X OX Y OY′ ′vkjS [khafp, tks x&v{k rFkk y&v{k dks

fu#fir djrh gSaA ,d gh LosQy ysdj x&v{k rFkk y-v{k ij fcanqvksa dks vafdr dhft,A

3. nh gqbZ nks izfrPNsnh js[kkvksa dks xzkI+kQ isij ij [khfp, rFkk mudk izfrPNsn fcanq] ekuk(h, k) dks vafdr dhft, (vko`Qfr 20.1 nsf[k,)

mís'; vko';d lkexzh

;g lR;kfir djuk fd a1x+b

1y+c

1=0 vkSj

a2x+b

2y+c

2=0 osQ izfrPNsnh fcanq ls tkus

okyh js[kk dk lehdj.k (a1x+b

1y+c

1) +

λ (a2x+b

2y+c

2) = 0 osQ izdkj dk gksrk gSA

dkMZcksMZ (xÙkk)] LosQp isu] lI+ksQn dkx”k]xksan] isafly] :yj

fØ;kdyki 20

04/26/2018


izn'kZu

1. eku yhft, fd js[kkvksa osQ lehdj.k 3x – 2y = 5 vkSj 3x + 2y = 7 gSaA

2. bu js[kkvksa dk izfrPNsnh fcanq 1

2,2

gS (vko`Qfr 20.2 nsf[k,)

3. nh gqbZ js[kkvksa osQ izfrPNsnh fcanq 1

2,2

ls tkus okyh js[kk dk lehdj.k gS%

( ) ( )3 – 2 – 5 3 2 – 7 0x y x y+ + =λ (1)

4.1

1, –1,2,2

λ = yhft,A

5. (i) 1,λ = osQ fy,] izfrPNsnh fcanq ls tkus okyh js[kk dk lehdj.k

(3x – 2y – 5) + (3x + 2y – 7), vFkkZr 6x – 12 = 0 gS tks izfrPNsnh fcanq 1

2,2

dks

larq"V djrk gS D;ksfd 6 (2) – 12 = 0

04/26/2018

xf.kr 65

(ii) –1λ = , osQ fy,] izfrPNsnh fcanq ls tkus okyh js[kk dk lehdj.k (3x – 2y – 5)

– 1 (3x + 2y – 7) vFkkZr~ – 4y + 2 = 0 gS] ;g Hkh izfrPNsnh fcanq 1

2,2

}kjk

larq"V gksrk gSA

(iii) 2λ = , osQ fy,] lehdj.k (3x – 2y – 5) + 2 (3x + 2y – 7) = 0, vFkkZr~

9x + 2y – 19 = 0, gS tks iqu% fcanq 1

2,2

}kjk larq"V gksrk gSA

izs{k.k

1. 3λ = osQ fy,] nh gqbZ js[kkvksa osQ izfrPNsnh fcanq ls tkus okyh js[kk dk lehdj.k

_________ gS tks fcanq 1

2,2

}kjk larq"V gksrk gSA

2. 4λ = osQ fy,] nh gqbZ js[kkvksa osQ izfrPNsnh fcanq ls tkus okyh js[kk dk lehdj.k________ gS tks nh gqbZ js[kkvksa izfrPNsnh fcanq ___________ }kjk larq"V gksrk gSA

3. 5λ = osQ fy,] nh gqbZ js[kkvksa osQ izfrPNsnh fcanq ls tkus okyh js[kk dk lehdj.k_____

gS tks nh gqbZ js[kkvksa osQ izfrPNsnh fcanq ________}kjk larq"V gksrk gSA

vuqiz;ksx

bl fØ;kdyki dk mi;ksx nks js[kkvksa osQ izfrPNsnh fcanq ls tkus okyh js[kk osQ lehdj.k lslacaf/r ifj.kkeksa dks le>us esa fd;k tk ldrk gSA ;g Hkh izsf{kr fd;k tkrk gS fd ,dfuf'pr fcanq ls vaurr% vusd js[kk,¡ tkrh gSaA

04/26/2018

lab manual xi (setting (hindi) on13-08-09) 11 20ncert.nic.in/ncerts/l/khlm402.pdf · tslk vko`qfr...

Documents